A characterization of \(K\)-invariant Stein domains in symmetric embeddings. (English) Zbl 0790.32030

Huckleberry, Alan T.; Fels, G.

Ancona, Vincenzo (ed.) et al., Complex analysis and geometry. New York: Plenum Press. The University Series in Mathematics. 223-234 (1993).

Let \(K\) be a connected compact Lie group and let \(G\) be its complexification. Let \(X\) be an irreducible normal Stein space equipped with a \(G\)-action. If \(G\) has an open orbit in \(X\) that is a complex symmetric space, then \(X\) is called an affine symmetric embedding. Each such \(X\) possesses an essentially canonical closed complex subvariety \(D\) that is an affine torus embedding with respect to a certain subgroup of \(G\) such that \(K\cdot D=X\).
The main result of the paper is the following theorem: Let \(\Omega\) be a connected \(K\)-invariant domain in an affine symmetric embedding \(X\). Then \(\Omega\) is Stein if and only if \(\Omega\cap D\) is Stein and connected.
For the entire collection see [Zbl 0772.00007].

Reviewer: A.Russakovskii (Khar’kov)

MSC:

32M05	Complex Lie groups, group actions on complex spaces
32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32E10	Stein spaces, Stein manifolds

Keywords:

\(K\)-invariant Stein domains; affine symmetric embedding